curl

If , show that . is a vector field and is a scalar field.
I know it's an obvious question but I can't remember how to formally prove it.
I know it's true if and in Cartesian coordinates, I've showed it for but not for a general form of . I think I'm over-complicating this.
Could you give me a tip?

If , show that . is a vector field and is a scalar field.
I know it's an obvious question but I can't remember how to formally prove it.
I know it's true if and in Cartesian coordinates, I've showed it for but not for a general form of . I think I'm over-complicating this.
Could you give me a tip?

It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...

It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...

I'm not expecting a particular kind of help so I appreciate yours.
I do not know how to relates Stokes' theorem with this problem.
I know that .
In the example given, I do not know where and are defined.
By intuition I could take as a circle with radius R tending to infinite and S being the upper half sphere of radius R but I'm not sure it makes sense nor how do I use . Am I, at least, on the right direction? If so I'll think more about it.

Edit: Now I'm thinking about using Clairaut's theorem. But I'd have to show that is continuous and it is not given...